Should mathematics be defined broadly as the study of mathematical structures
or
should it be defined narrowly as production of proofs in formal systems with
recursively enumerable provability relation?
For example, a solution to the continuum hypothesis would involve producing
results in particular formal systems (which is mathematics), and giving a
highly persuasive argument that the formal results (which are Sigma-0-1
statements) imply the continuum hypothesis (or its negation), and the question
is whether that persuasive argument should be considered to be mathematics as
well.
Dictionaries generally use the broader definition, but that may be because the
narrower definition is too technical, and knowledge about mathematical
structures is ordinarily obtained by proofs from self-evident axioms. I think
that mathematical journals should sometimes publish philosophical and empirical
arguments about truthfulness of mathematical statements. However, unless
stated or clearly understood otherwise, a claim that a mathematical statement
is true includes and should continue to include an implicit claim that enough
is known so that it is "routine" to produce a formal proof of the statement in
the appropriate formal system (typically, ZFC).
Dmytro Taranovsky